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Fracture in Composite Laminates
András Szekrényes, József Uj
To cite this version:
Mixed-mode I/II Interlaminar
Fracture in Composite Laminates
ANDRA´SSZEKRE´NYES* ANDJO´ZSEFUJ
Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, PoB 11, H-1521, Hungary
ABSTRACT: In this work the over-leg bending (OLB) specimen is developed for mixed-mode I/II delamination characterization in composites. The traditional single-leg bending (SLB) specimen is modified by introducing the load eccentrically between the two supports of a three-point bending setup. The modified configuration is analyzed by using linear beam theories. The theories of transverse shear, Winkler-Pasternak-type elastic foundation, Saint-Venant effect, and crack tip shear deformation are incorporated in the analysis. Also, experiments are carried out on glass/polyester unidirectional specimens. Comparison between the results of analysis and experiment shows very good agreement. The traditional SLB and the novel OLB coupons are compared, and their advantages and drawbacks are highlighted. The newly developed OLB setup is relatively easier to perform and crack propagation may be simply investigated under displacement control.
KEY WORDS: composite delamination, beam theory, mode-mixity, fracture, SLB specimen, OLB specimen.
INTRODUCTION
T
HE INTERLAMINAR FRACTUREtests play an important role in the designand development of composite structures. Usually, these tests are performed by using specimens, which may be considered as slender beams. The double-cantilever beam (DCB) (Olsson, 1992; Yang and Sun, 2000; Morais et al., 2002) specimen is a standard tool to measure the mode-I fracture in composite materials.
International Journal ofDAMAGEMECHANICS, Vol. 16—January 2007 5
1056-7895/07/01 0005–29 $10.00/0 DOI: 10.1177/1056789507060774
ß 2007 SAGE Publications
For mode-II testing, six specimens are available. The end-notched flexure (ENF) (Carlsson et al., 1986; Ozdil and Carlsson, 1998) is the subject in most of the works. In contrast, its stabilized form (SENF) was discussed only in a few works (e.g., Davies et al., 1996). The four-point bend end-notched flexure (4ENF) test is also a popular geometry (Schuecker and Davidson, 2000; Davies et al., 2004). Another candidate for mode-II fracture investigation is the end-loaded split (ELS) coupon (Hashemi et al., 1990a, b; Wang and Vu-Khanh, 1996). The over-notched flexure (ONF) specimen (Tanaka et al., 1998; Wang et al., 2003; Szekre´nyes and Uj, 2005b) is the modified version of the ENF one. The only difference between them is that in the ONF setup the load is introduced eccentrically between the two supports of the three-point bending apparatus. More recently, the tapered end-notched flexure (TENF) specimen was proposed by Wang and Qiao (2003) and Qiao et al. (2004). A remarkable feature is that in the case of a proper design the compliance rate change is independent of the crack length, which is useful, for instance in the case of carbon-fiber reinforced systems, when the crack length is difficult to measure accurately. All these specimens have advantages and relative drawbacks (Davies et al., 1996; Szekre´nyes and Uj, 2005b).
Davidson et al. performed extensive theoretical and experimental works using SLB coupons (Davidson et al., 1995, 1996, 1997, 1999, 2000). This type of specimen was also investigated by Polaha et al. (1996) and Pieracci et al. (1998). Later, the SLB test was modified by Tracy et al. and the single-leg four point bend (SLFPB) geometry was introduced (Tracy et al., 2003). In the current study we developed another alternative: the over-leg bending (OLB) specimen.
In general the fracture specimens are treated as slender beams. There are several analytical solutions, which are based on beam and plate models. The solution by Reeder and Crews (1990), or Ozdil and Carlsson (1999) for the MMB specimen and Olsson for the DCB specimen (Olsson, 1992) are based on the elastic foundation model by Williams (1989). There were many efforts to derive a correction also for mode-II specimens. See for instance the works by Carlsson et al. (1986), Chatterje (1991), and Ding and Kortschot (1999). None of these solutions were satisfactory. This void was addressed by Wang and Qiao (2004a), who presented an improved solution for the mode-II ENF specimen, which had a similar form to that of the elastic foundation model by Williams (1989).
and the crack propagation is easy to control. Furthermore, a simple reduction technique can be applied to reduce the experimental data.
BEAM THEORY-BASED ANALYSIS
The SLB and OLB mixed-mode I/II delamination coupons are shown in Figure 1. The fracture properties of the specimens are calculated based on previous works. The expressions incorporate transverse shear analysis (Ozdil and Carlsson, 1998), Saint-Venant (SV) effect (Olsson, 1992), Winkler-Pasternak-type elastic foundation (Szekre´nyes and Uj, 2004, 2005c), and crack tip shear deformation analysis (Wang and Qiao, 2004a). For the sake of completeness, we give a brief summary on the applied models and theories. Usually, the application of the Euler–Bernoulli (EB) and Timoshenko (TIM) beam theories are not too complicated, and for the common fracture specimens the compliance and strain energy release rate (SERR) expressions are known. Consequently, the compliance and strain energy release rate may be obtained in a fashion similar to that shown for instance in Ozdil and Carlsson (1998) or Szekre´nyes and Uj (2004). The compliance and the strain energy release rate of the SLB specimen shown in Figure 1(a) is (Szekre´nyes and Uj, 2004):
CSLB¼CSLBEB þCSLBTIM¼7a 3þ2L3 8bh3E11 þ a þ2L 8bhkG13 , ð1Þ (a) (b) P(2L−s) 2L Ps 2L P h h a s 2L 2 P 2 P P h h a L L
where the first term is the result of the EB beam theory and the second one is from TIM beam theory. The strain energy release rate may be obtained by using the Irwin-Kies expression (e.g., Olsson, 1992):
GI=II¼ P2 2b dC da, ð2Þ i.e., we have: GSLBI=II ¼ 21P 2a2 16b2h3E 11 þ P 2a2 16b2h3E 11 fT, ð3Þ where fT¼ 1 k E11 G13 h a 2 : ð4Þ
The compliance of the OLB specimen can be obtained by performing a similar analysis (incorporating E-B and TIM beam theory) to that in Ozdil and Carlsson (1998) or Szekre´nyes and Uj (2004):
COLB¼CEBOLBþCTIMOLB
¼ s 2c3 8bh3E 11L2 1 þ 8a cþ16 aL c2 þ32 aL2 c3 þ16 Ls s ð 4LÞ c3 þs½4L 2L sð Þ 8bhkG13L2 : ð5Þ
The SERR can be calculated combining Equations (2) and (5):
GOLBI=II ¼ 21P 2s2c2 16b2h3E 11L2 þ P 2s2c2 16b2h3E 11L2 fT: ð6Þ
In Equations (1)–(6) a is the crack length, L is the half span length, b is the specimen width, h is half of the specimen thickness, c ¼ 2L a is the length of the uncracked region, k ¼ 5/6 is the shear correction factor, E11 is the
flexural modulus, E33 is the through-thickness modulus, G13 is the shear
Winkler-Pasternak Foundation Analysis
The effect of the two-parameter elastic foundation was studied in our previous work (Szekre´nyes and Uj, 2005c). Here we summarize the result of the analysis. Let us consider the general loading scheme depicted in Figure 2. The model has four deflection functions. Two of them are EB beams (cracked region), while in the uncracked portion the beam is mounted on an elastic foundation. The analysis was performed according to the following:
. The governing differential equations of the uncracked portion were formulated by using the variational formulation of the potential energy. . The coefficients in the differential equations were determined and were
related to the elastic properties of the material.
. The solution functions of the uncracked region were found in the form of the combination of hyperbolic functions, the cracked region was modeled as EB beams.
. The continuity and boundary conditions were formulated and the coefficients of the deflection functions were determined.
. The compliance of both arms in the model was determined by taking the deflection at the point of load applications.
The solution resulted in the following expressions for the compliances of the upper (1) and lower (2) arms:
CWP1¼ 7a3þL3 2bh3E 11 þ L 3a3 P2 2bh3E 11P1 þa 3 P 1P2 ð Þ 2bh3E 11P1 fW1, ð7Þ CWP2¼ 7a3þL3 2bh3E 11 þ L 3a3 P1 2bh3E 11P2 þa 3 P 1P2 ð Þ 2bh3E 11P2 fW1, ð8Þ h h P1 ke kG wl1(x) wl2(x) wu1(x) P2 a c wu2(x) x z
where fW1¼2:711=2 h a E11 E33 1=4 þ2:45 h a 2 E11 E33 1=2 þ1:111=2 h a 3 E11 E33 3=4 , ð9Þ
where ¼ 1 þ !, and ! is a parameter related to the Pasternak-type foundation. A series of FE analyses on composite specimens with different material properties resulted in ! ¼ 2.5, which may be considered as a constant (Szekre´nyes and Uj, 2005c). The first two terms in Equations (7) and (8) are the result of the EB beam theory, while the third one is the correction based on the two-parameter elastic foundation. The energy release rates for the upper and lower arms may be obtained by using Equation (2): G1¼ P2 1 2b dC1 da , G2¼ P2 2 2b dC2 da : ð10Þ
Thus, we may obtain:
GWP1þGWP2¼ 21 P2 1þP22 a2 4b2h3E 11 6P1P2a 2 4b2h3E 11 þðP1P2Þ 2 a2 4b2h3E 11 fW2, ð11Þ where, fW2¼5:421=2 h a E11 E33 1=4 þ2:45 h a 2 E11 E33 1=2 ð12Þ
Note that the result of the Winkler foundation model may be recovered by using ! ¼ 0 in Equations (9) and (12) (Szekre´nyes and Uj, 2004).
Saint-Venant Effect at the Crack Tip
where is the characteristic decay length in the material. This means that the strain energy related to the SV effect decays exponentially ahead of the crack tip. A relevant analysis resulted in the following compliance contributions from the upper and lower arms (Szekre´nyes and Uj, 2005c):
CSV1¼CSV2¼ 6 a2 bh2E 11 E11 G13 1=2 , ð14Þ
It has been shown that the energy release rate caused by the Saint-Venant effect may be written as (Szekre´nyes and Uj, 2005c):
GSV¼ P1P2 ð Þ2a2 4b2h3E 11 fSV, ð15Þ where, fSV¼ 12 h a E11 G13 1=2 : ð16Þ
Crack Tip Shear Deformation Analysis
The concept of crack tip shear deformation was originally provided by Wang and Qiao (2004a). Their solution was adopted in our last works
θ θ M 2h x z θ θ
(Szekre´nyes and Uj, 2005b, 2005c), wherein the same concept was used to obtain the solution of the general model depicted in Figure 4. Again, we give only a short description on the results obtained:
. The shear stress field ( (x)) and the deflections of the beam in Figure 4 were determined by following the way suggested by Wang and Qiao (2004a)
. The continuity and boundary conditions were also formulated, the coefficients of the deflection functions were determined
. The compliance of both arms of the model was determined by taking the deflection at the point of load application.
After some simplifications the analysis resulted in the following compliance contributions: CSH1¼ wu1ðaÞ P1 ¼ðP1þP2Þa 3 2P1bh3E11 fSH1, ð17Þ CSH2¼ wl1ðaÞ P2 ¼ðP1þP2Þa 3 2P2bh3E11 fSH1, ð18Þ where, fSH1¼0:98 h a E11 G13 1=2 þ0:43 h a 2 E11 G13 : ð19Þ
The SERR for the upper and lower arms by using Equation (10) are:
GSH1þGSH2¼ P1þP2 ð Þ2a3 4b2h3E 11 fSH2, ð20Þ P2 P1 wu2(x) wl2(x) wl1(x) wu1(x) h h τ02 τ01 a c x z τ(x)
where, fSH2¼1:96 h a E11 G13 1=2 þ0:43 h a 2 E11 G13 : ð21Þ Mode-mixity Analysis
According to Figure 5 the authors reduce problem (a) into problem (b), wherein the effect of Winkler-Pasternak foundation, transverse shear, SV and crack tip shear deformation are incorporated. In Figure 5 M1¼P1a,
M2¼P2a, and M3¼M1þM2 are bending moments at the crack tip. The
sum of Equations (11), (15), and (20) can be transformed as:
GT¼ 21 M2 1þM22 6M1M2þðM1M2Þ2ðfW2þfSVÞ þðM1þM2Þ2fSH2 4b2h3E 11 : ð22Þ
The authors assume that the equivalent bending moments can be decomposed as follows (Williams, 1988):
M1¼MIþMII, M2¼MIþ’MII: ð23Þ h h P1 P P P2 a c x z h h M1 a c x z M3 M2 (b) (a)
Substituting Equation (23) into Equation (22) the total SERR are obtained in the form of:
GT¼GIþGIIþGIþII, ð24Þ
where GI and GII are the individual mode-I and mode-II SERR
components, G
IþII is a term, which contains the product of MIMII.
The last term should be eliminated. Williams (1988) supposes that in the case of pure mode-II the upper and lower specimen arms have the same curvature: 12MII bh3E 11 ¼12’MII bh3E 11 : ð25Þ
From this beam kinematics assumption it follows that ’ ¼ 1. Substituting Equation (23) into Equation (22) the term containing the product of MIMII
can be separated:
GIþII¼ð9 þ fSH2ÞMIMIIð1 þ Þ b2h3E
11
: ð26Þ
In order to cancel Equation (26) we choose ¼ 1. Referring to (Szekre´nyes and Uj, 2004), wherein it has been found that the transverse shear ( fT) contributes only to the mode-I component,
and from Equations (22) and (23), the mode-I and mode-II components become: GI¼ M2 Ið12 þ fW2þfTþfSVÞ b2h3E 11 , ð27Þ GII¼ M2 IIð9 þ fSH2Þ b2h3E 11 : ð28Þ
Rearranging Equation (23) the mode-I and mode-II bending moments are obtained as follows: MI¼ M1M2 2 , MII¼ M1þM2 2 : ð29Þ
where fW2 is given by Equation (12), fT is given by Equation (4), fSV is
Compliance Calculation
The authors give a general expression, which can be applied to calculate the compliance for a beam under general loading condition. In a general form the mode-I and mode-II bending moments (Equation (29)) may be written as:
MI¼P fIa, MII¼P fIIa, ð30Þ
where P is the external load, fIand fIIare related to geometrical parameters
of the system and a* is the characteristic length (in each case may be related to the crack length). As it was shown, the two-parameter elastic foundation and SV effect contribute only to the mode-I component, while the crack tip shear deformation improves only the mode-II SERR. Therefore, in a general form the compliance of delaminated beams may be obtained as:
C ¼ CEBþCTIMþ f2 Ia3 2bh3E 11 fSV 2 þfW1 þ f 2 IIa3 2bh3E 11 fSH1, ð31Þ
where the determination of the terms CEB and CTIM requires a unique
analysis. Moreover, fW1 is given by Equation (9), fSV is defined by
Equation (16) and fSH1 is defined by Equation (19). The application of
Equations (27), (28), and (31) will be demonstrated in the next section.
Application to the SLB and OLB Specimens
Let us consider the case of the SLB specimen in Figure 1(a), where M1¼Pa/2 and M2¼0. From Equation (29) it follows that MI¼MII¼Pa/4.
Furthermore, comparing Equation (29) to (30) we may establish that fI¼fII¼1/4 and a* ¼ a. Substituting these into Equation (31) and also using
Equation (1), the following expression for the compliance of the SLB specimen are obtained:
where the first two terms are given by Equation (1), the third is from the crack tip shear deformation, the fourth captures the SV effect, while the last term is the effect of the Winkler-Pasternak foundation. Also, using Equations (27) and (28) with the aid of the aforementioned moments (MI, MII), the following is obtained:
GSLBI ¼ 12P 2a2 16b2h3E 11 1 þ 0:85 h a E11 E33 1=4 þ0:71 h a 2 E11 E33 1=2 " þ0:32 h a E11 G13 1=2 þ0:1 h a 2 E11 G13 # , ð33Þ GSLBII ¼ 9P 2a2 16b2h3E 11 1 þ 0:22 h a E11 G13 1=2 þ0:048 h a 2 E11 G13 " # : ð34Þ
In what follows, the compliance and the energy release rate of the OLB specimen are calculated. In this case it may be written that M1¼Ps(2L a)/
2L and M2¼0 (Figure 1(b)). From Equation (29) it follows that MI¼MII¼
Ps(2L a)/4L. Comparing Equations (29) and (30) that it is concluded fI¼fII¼s/4L and a* ¼ 2L a. This means that the crack length ‘a’ in
Equations (4), (9), (12), (16), (19) and (21) should be replaced with 2L a. Substituting 2L a with c in Equation (31) and also incorporating Equation (4) the following is obtained:
The energy release rate components by the help of Equations (27) and (28) are: GOLBI ¼ 12P 2s2c2 16b2h3E 11L2 1 þ 0:85 h c E11 E33 1=4 þ0:71 h c 2 E11 E33 1=2 " þ0:32 h c E11 G13 1=2 þ0:1 h c 2 E11 G13 # , ð36Þ GOLBII ¼ 9P 2s2c2 16b2h3E 11L2 1 þ 0:22 h c E11 G13 1=2 þ0:048 h c 2 E11 G13 " # , ð37Þ
where s is the position of the applied load from the left side of the specimen and c ¼ 2L a is the length of the uncracked part. The total strain energy release rate (GI/II¼GIþGII) can be calculated by summing
Equations (33)–(34) and (36)–(37). The mixed mode-ratio (GI/GII) can be
also computed for both specimens.
FINITE ELEMENT ANALYSIS
The FE analysis was performed using the commercial COSMOS/M 2.0 finite element package (Structural Research and Analysis Corp., 1996) to validate the beam theory-based compliance expressions. The models of the SLB and OLB specimens were built by using linear elastic PLANE2D elements under a plane strain condition. Note that the beam formulation of the problems involves a plane stress condition. The boundary conditions were the same as those illustrated in Figure 1. The specimens were loaded by a concentrated force equal to unity and the compliance of the specimens was determined in the extended ranges of the crack length (for details see the next section). The mode–mix ratio (GI/GII) was
EXPERIMENTS AND DATA REDUCTION
The constituent materials were procured from Novia Ltd. The properties of the E-glass fiber are E ¼ 70 GPa and ¼ 0.27, while for the unsaturated polyester resin are E ¼ 3.5 GPa and ¼ 0.35. Both were considered isotropic. The unidirectional ([0]
14) glass/polyester specimens with a nominal
thickness of 2h ¼ 6 mm, a width of b ¼ 20 mm, and a fiber-volume fraction of Vf¼43% were manufactured in a special pressure tool. A great
advantage of this material is the transparency, which allows for the visual observation of crack initiation/propagation. A nylon insert with a thickness of 0.03 mm was placed at the midplane of the specimens to make an artificial starting defect. Then the specimens were precracked in an opening mode of 4–5 mm. The flexural modulus was determined from a three-point bending test using six uncracked specimens. The experiment resulted in E11¼33 GPa, additional properties were predicted by using simple rule of
mixture, this way E33¼7.2 GPa, G13¼3 GPa and 13¼0.27 were obtained.
Crack initiation tests were carried out for both the SLB (Figure 7(a)) and OLB (Figure 7(b)) specimens. In the case of the SLB test the full span length was 2L ¼ 151 mm. The specimens with initial crack lengths from 20 to 75 mm with 5 mm increment were prepared. For the OLB test the span length was also 2L ¼ 151 mm. Furthermore, the position of the external load
Crack
0.5 mm
was s ¼ 47.5 mm. In this case the crack length ranges from 55 to 115 mm with 5 mm increment was investigated, and crack propagation tests using six specimens with a ¼ 50 mm initial crack length were also performed. The tests were carried out using an Amsler testing machine under displacement control. The load–deflection data was measured. The latter was monitored by the dial gauge, shown in Figure 7. The crack length was measured visually. For this reason a millimeter scale was traced on the lateral sides of the specimens and the position of the crack tip was marked on the upper surface of the specimens. The contact regions above the supports were slightly roughened in order to prevent longitudinal sliding.
Two methods were used for data reduction: beam theory-based approach described in the ‘beam theory-based analysis’ section 2 and the compliance calibration (CC) (e.g., Schuecker and Davidson, 2000) method. In the case of the SLB test the experimental compliance values were fitted by the following polynomial expression (see e.g., Szekre´nyes and Uj, 2004):
CSLB¼C01þma3, ð38Þ
where C01and m were found by using least square fitting. The compliance of
the OLB specimen may be found in a similar form to that of the ONF specimen (Wang et al., 2003):
COLB¼C02þn a ð 2LÞ3: ð39Þ
The coefficients C02 and n were determined by least square fitting. The
SERR may be calculated using Equation (2).
Dial gauge measuring the specimen displacement (b) OLB (a) SLB Support Specimen Roller
RESULTS AND DISCUSSION
Crack Initiation Tests
Figure 8 shows the load–displacement curves recorded up to fracture initiation for both specimens. Each curve is typically linear, but the two tests produce quite distinct results. For the OLB test both the critical load and displacement increase with crack length. The ranges of the applied load are essentially the same in both tests, but it should be kept in mind that the crack length ranges are different. The SLB test indicates that the critical load reaches the highest value if the crack length exhibits the lowest value
(b) (a) a=30 mm Load - P (N) a=40 mm a=50 mm a=60 mm 0 1 2 3 4 5 a=70 mm 0 150 300 450 600 a=20 mm Displacement -d (mm) 0 4 8 12 16 20 a=60 mm a=70 mm a=80 mm a=90 mm a=100 mm 0 150 300 450 600 a=110 mm Load - P (N) Displacement - d (mm)
(Figure 8(a)). The results of the OLB are in sharp contrast with these facts, i.e., the critical load increases with the initial crack length.
In Figure 9 we demonstrate the deformation of the FE mesh of the OLB specimen. The displaced shape of the SLB specimen is similar. The mesh has a high density, therefore a detailed view is included at the crack tip. The compliances calculated from the beam theory-based solution were compared with the results of the FE analysis. In each case the compliance values are normalized with the result of the EB beam theory. These are the first terms in Equations (1) and (5). Tables 1 and 2 list the normalized values of the two solutions for both configurations. Equations (32) and (35) show very good agreement with the FE results, which confirms the applicability of the analytical solution. The mode–mix ratio is also collected in Tables 1 and 2. The ratio by the FE analysis changes within 1.63–1.32 in the case of the SLB and decreases from 1.32 to 1.24 in the case of the OLB specimen. For the
P
Crack tip
Figure 9. Deformation of the finite element mesh, scale factor: 0.7.
Table 1. Comparison of the results by the FE and beam models, SLB specimen.
a (mm) 20 25 30 35 40 45 50 55 60 65 70 75
CFE/CE–B 1.058 1.071 1.084 1.096 1.105 1.112 1.115 1.117 1.116 1.115 1.112 1.105
CBeam/CE–B 1.051 1.062 1.073 1.082 1.089 1.094 1.096 1.097 1.096 1.094 1.091 1.088
(GI/GII)Beam 1.675 1.604 1.557 1.524 1.499 1.480 1.465 1.453 1.443 1.434 1.427 1.420
(GI/GII)FE 1.628 1.579 1.545 1.520 1.503 1.485 1.476 1.468 1.462 1.449 1.457 1.324
SLB specimen the solution based on beam theory predicts values between 1.68 and 1.42 as the crack length increases. This is in excellent agreement with the VCCT method. On the other hand in the case of the OLB specimen the mode ratio by beam analysis has values between 1.4 and 1.52 with the increasing crack length. According to the VCCT method the crack length dependency of the mode–mix ratio is somewhat stronger in the case of the SLB coupon.
In order to explain the disagreement between the result of the VCCT and the beam model for the mode ratio of the OLB specimen, we performed FE analyses in which the position of the applied load was varied. Note that the crack length was held at a constant value (a ¼ 60 mm). The result of the analysis is listed in Table 3. It transpires that the mode ratio is approximately constant until the position of the load reaches the crack tip. At this point it seems that there is a little transition, where the mode ratio slightly enhances. If the position of the load is higher than the position of the crack tip there is a jump in the mode ratio. It is apparent that this effect is not included in the beam model, which (assuming similar behavior at the other crack lengths) may explain the disagreement with respect to the mode ratios.
The compliance curves determined for the SLB and OLB coupons based on crack initiation tests are illustrated in Figures 10(a) and 11(a). The advanced beam model (Equation (32)) of the SLB coupon slightly overpredicts the experimental points, although the overpredictions are not significant. The slightly unusual compliance curves were calculated based on experimental data from the OLB specimens. In this case the characteristic distance c, in the beam theory solution (Equations (35), (36), and (37)), is the length of the uncracked region. In spite of this, the agreement between the result of the beam theory-based solution (Equation (35)) and the experiment is excellent. The values of the SERR at propagation onset are plotted in Figures 10(b) and 11(b). In the case of the SLB specimens the CC method results in a 645 J/m2plateau value, while the beam model predicts a 625 J/m2 for the steady-state value. The results show opposite trends in the case of the OLB specimen, i.e., the beam model overpredicts the experiments. The CC method reports a 650 J/m2 plateau value, while
Table 3. The effect of the position of the applied load on the mode ratio (a ¼ 60 mm).
s (mm) 20 30 40 50 60 70 80 90 100
(GI/GII)FE 1.314 1.320 1.314 1.317 1.344 1.467 1.457 1.454 1.461
the relevant result of the beam model is 715 J/m2 (9% difference). It should be mentioned that less than 1% difference was found between the steady-state SERR values of the SLB and OLB configurations (645 J/m2 against 650 J/m2). Consequently, the two tests lead to essentially the same result.
In our previous work using the same material the SCB setup also resulted in a 645 J/m2 steady-state SERR value. Similar trends were also obtained by others using MMF specimens. The only difference between the MMF and SLB configurations is the distinct thicknesses above the left support. Albertsen et al. (1995) reported GI/II, init¼145 J/m2initiation and
(b) (a) 0 8 16 24 32 R2=0.9986 Experiment
Third order polynomial fit Beam theory, Eq. (32)
Compliance - C (mm/N) × 10 − 3 Crack length - a (mm) 0 16 32 48 64 80 0 16 32 48 64 80 0 140 280 420 560 700 a+bx+cx2+dx4 625 J/m2 645 J/m2 Polynomial fit(1) Polynomial fit(2) Polynomial fit(3) Polynomial fit(4) GI/IIC - CC method - (1)
GI/II - Beam theory - (2) GI - Eq. (33) - (3) GII - Eq. (34) - (4)
Strain energy release rate - G
I/II
(J/m
2)
Crack length - a (mm)
GI/II, ss¼330 J/m2 mean propagation values for carbon-fiber reinforced
MMF specimens. In the work of Korjakin et al. (1998) the relevant values were GI/II, init¼456 J/m2 and GI/II, ss¼525 J/m2 for glass/epoxy MMF
specimens. The character of the SERR followed the same trend as is found in Figures 10(a) and 11(a).
The values of the critical load and critical displacements up to propagation onset (or crack initiation) are illustrated in Figure 12(a). An immediate observation is that the load values recorded during the SLB test follow a hyperbolic character, while the load values from the OLB test increase with crack length. The critical displacement (Figure 12(b)) in
(b) (a) 40 56 72 88 104 120 0 190 380 570 760 a+bx+cx2+dx3 650 J/m2 715 J/m2 Polynomial fit(1) Polynomial fit(2) Polynomial fit(3) Polynomial fit(4) GI/IIC - CC method - (1)
GI/II, Beam theory - (2) GI, Eq. (36) - (3) GII, Eq. (37) - (4)
Strain energy release rate - G
I/II (J/m 2) Crack length - a (mm) 50 64 78 92 106 120 10 16 22 28 34 R2=0.9885 Experiment
Third order polynomial fit Beam theory, Eq. (35)
Compliance - C (mm/N) × 10 − 3 Crack length - a (mm)
the SLB test reaches its lowest value at a ¼ 40 mm. On the other hand the displacement increases dramatically if the results of the OLB test are considered. The measured load and displacement values against the crack length in each case may fit very well by using third-and fourth-order polynomials, as shown in Figure 12(a) and 12(b).
(b) (a) OLB specimen a+bx+cx2+dx3+ex4, R2=0.9993 0 24 48 72 96 120 0 225 450 675 900 SLB specimen a+bx+cx2+dx3, R2=0.9996 Critical load (N) Crack length - a (mm) 0 24 48 72 96 120 Crack length - a (mm) OLB specimen ax+bx+cx2+dx3+ex4, R2=0.9986) 0 6 12 18 24 SLB specimen ax+bx+cx2+dx3, R2=0.9928 Critical displacement (mm)
Crack Propagation Tests
Under the current geometrical parameters the SLB specimens were not suitable for propagation tests. At shorter crack lengths (a ¼ 20 to 45 mm) sudden jumps (15 to 30 mm) in the crack advance frequently occurred. The remaining interval (until the point of load introduction) was insufficient for studying crack propagation. Therefore, the OLB test was used as a candidate to examine mixed-mode crack propagation. The compliance curves of each six specimens were essentially the same as those determined through the initiation test (Figure 11(a)). The load–displacement curves of six specimens are depicted in Figure 13(a). The range of the applied load is eventually the same as those which were recorded during initiation tests (Figure 8(b)). The crack initiation always occurred at P ¼ 170 to 180 N.
The propagation R-curves are plotted in Figure 13(b) and (c). Although six specimens were tested, the results of only two specimens are displayed. The curves show the R-curve behavior, i.e., there is not a clear plateau value. Overall the SERR grows up to 1250 J/m2. The agreement between the result of the CC method and beam theory is also good. Note that fiber-bridging was observed during testing, although this effect was not considered here. Furthermore, due to the relatively small displacements before the crack tip region, the significance of this phenomen was estimated to be small. The extension of the fiber-bridging was only mild in each specimen. The tested specimens at the end of the delamination process are shown in Figure 14. The initially curved crack front became curved after crack propagation onset.
For comparison, Tracy et al. (2003) determined similar propagation R-curves with 1500 and 1750 J/m2 plateau values using the proposed SFPLB configuration with carbon/epoxy specimens to those, shown in Figure 13. Beam theory-based equations were used for data reduction in their study.
CONCLUSIONS
(c) (b) (a) 0 5 10 15 20 25 0 200 400 600 800 Specimen 1 Specimen 2 Specimen 3 Specimen 4 Specimen 5 a0= 50 mm Crack initiation Specimen 6 Load (N) Displacement - d (mm) 40 54 68 82 96 110 Specimen 1 Fit1: b*log(x-a)
Strain energy release rate - G
I/II (J/m 2) Crack length - a (mm) Specimen 3 Fit3: b*log(x-a) 40 54 68 82 96 110 0 Specimen 3 Fit3: b*log(x-a) Specimen 1 Fit1: b*log(x-a)
Strain energy release rate - G
I/II (J/m 2) Crack length - a (mm) 1400 0 350 700 1050 1400 1050 700 350
The crack initiation tests were performed utilizing the SLB and OLB setups. In both cases quite good agreement was established between the experimentally and analytically determined compliance values. Considering the steady-state value of the SERR at crack initiation, the two tests produce values within 1% difference (645 J/m2 against 650 J/m2). The R-curves (under crack propagation) were determined using the new configuration based on two reduction schemes, and good agreement was found. This indicates that the used beam model can be applied for data reduction.
In the OLB test the large displacements do not influence the crack propagation and the propagation can be easily controlled. Moreover, the test essentially gives linear elastic response, and simple reduction technique (CC method) applies for data evaluation. The OLB coupon promotes more stable crack propagation than the SLB due to the possibility of greater crack length. A relative drawback of the OLB setup is that the mode ratio may be changed (similarly to the SLB and SCB coupons) only in a small degree.
ACKNOWLEDGMENTS
This research work was supported by the Hungarian Scientific Research Fund (OTKA) under Grant No. T037324. We wish to thank, Tonny Nyman for providing the references by Yoon and Hong (1990) and Williams (1988); Barry D. Davidson for providing the following references: Davidson and Sundararaman, 1996; Davidson et al., 1997, 2000; Pieracci et al., 1998 and Polaha et al., 1996); Pizhong Qiao for the Ref. by Wang and Qiao (2003), Alfredo Balaco´ de Morais for (Davies et al., 1996); and Tong-Earn Tay for providing his work (Tay, 2003). The first author is grateful to his father for manufacturing the experimental tools.
Crack front Direction of crack propagation
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